When the killer shoots the man at the beginning of the episode there was very little recoil from the gun. This gun, which is used to stop elephants, would have a tremendous amount of recoil.
When Monk and Stottlemeyer bring the elephant into the tent to confront Natasha it has on its head dress or costume but when she tries to escape and is about to run over Sharona an elephant steps in front of her. The elephant was no longer wearing anything.
The gun used by Natasia at the beginning of the episode is referred to a "454 Ruger Casull", however this is incorrect. While Ruger does make a .454 Casull handgun, it's a double action revolver, the one used is clearly a single action revolver, and only Freedom Arms and Magnum Research make a single action revolver in .454 Casull.
In the scene where Sharona is sitting in the ring, having been upset by the elephant and Monk is doing a really bad job of comforting her, he eventually comes to sit by her. Given his pathological fear of germs and his quest for cleanliness, it seems very strange that he would just sit down next to her, without at least trying to clean the seat, especially as we've seen him cleaning other seats that are much less likely to be dirty.
Natasia would not have left the walkie talkie behind the elephant's ear to be found. She's obviously smart enough to plot an elaborate murder. Why would she just abandon the murder "weapon" for anyone to find, knowing that Monk is already suspicious of her?
The doctor who x-rayed Natasia's foot could have determined that it had been broken less than two weeks by the degree of subsequent healing that had taken place. It would not have healed enough in just four days to look like a two-week old break. Also, a break caused by extreme pressure to the side of the ankle would have been very different from a break caused by the fall that she claims caused the break. The doctor would have recognized this discrepancy.
When Sharona lit her cigarette in the hospital, Monk pointed out that you can't smoke in hospitals. As a police officer, shouldn't Stottlemeyer have made her put it out?
While the elephant trainer was showing off the elephant's abilities to Monk and Sharona, Monk went to stand on the elephant's right side--the same side where the walkie-talkie was planted. With Monk's sharp eye for detail, he would have seen the walkie-talkie when the elephant flipped her ear forward.
Floppy the Clown: Squawk!
Monk: What's your problem?
Stottlemeyer: Hey, clown. Take a walk.
Sharona: S.I.U. Know what that means?
Monk: Yes, I know what that means.
Monk: How do you feel?
Sharona: Good. I think I'm over it. Maybe there's hope for you.
Monk: I'm sensitive. I feel your pain. I'm feeling your pain.
Sharona: I've created a monster.
Monk: I hear that. I'm hearing what you're saying.
Monk: How many pinches to a dollop?
Monk: Is this a dollop?
Monk: A dollop. It says add one dollop of whipped cream.
Monk: (to Sharona) We just thought that if you met the elephant, if you got to know her, you might get over your fear.
Monk: The stuff I'm afraid of is based on fact.
Sharona: It is?
Monk: Like germs.
Sharona: Like crooked paintings and sidewalk cracks and, and food touching. And the wind?
Human Cannonball: Maybe I should come to your office and rip through your stuff, huh?
Disher: (aside to Stottlemeyer) I hate that cannonball.
Dr. Kroger: Adrian, the answer is inside you.
Monk: No, doctor. The answer is inside you. If you told me, I would hear it, and then the answer would be inside me.
Monk: How did you know they were from me?
Sharona: All the same height. All the flowers have the same number of petals on them.
Monk: (on the phone) It's me, Adrian. Adrian Monk. Your boss.
Sharona: You're the most selfish, inconsiderate man I've ever seen.
Monk: Great. Now I'm upset.
Monk: (to Sharona) Tell the captain to meet us here. She's the guy.
Sharona: You promise you won't laugh?
Monk: I never laugh.
Ariana: How did you know that?
Monk: That pamphlet in your bag. You're studying the U.S. Constitution, something no U.S. citizen would do.
Stottlemeyer: (to Officer Myers) I couldn't care less about your [mismatched] socks. It's Monk. You're distracting him.
Officer Myers: That's Sharona, his nurse.
Disher: He has a nurse?
Sharona: We're not married. And if we ever get married--shoot me.
Monk: You know who you should marry? The elephant man.
Sharona: I'd marry the elephant man before I'd marry you.
Sharona: Do you kiss your mom with that mouth?
Sharona: Well, maybe you should. Come here.
Benjy: I love you.
Sharona: I love you, too.
Monk: Hello. I'm Adrian Monk and this is my assistant Sharona. She'll say "hello" later.
The circus, Dratch and Denby, is named after one of the show's writers, Daniel Dratch ("Mr. Monk and the Very, Very Old Man"), who was the story editor for this episode.
Monk: You know who you should marry? The Elephant Man!
Referencing the Elephant Man, the stage name given to Joseph Carey Merrick, a Victorian era performer in Britain born with very severe physical deformities. He was exhibited as a human curiosity called "The Elephant Man."